Why Windshield Wipers Are Moving Geometry
A windshield wiper seems to perform one simple task: sweep water away from a driver's view. Underneath that motion is a rich mathematical system. A blade rotates through an angle, traces an annular sector, moves at different linear speeds along its length, and follows a linkage whose output is not necessarily uniform even when the motor turns steadily. Two blades create overlapping swept regions. Tolerances, parking positions and cycle rates add measurement and timing.
This article uses simplified geometry to explain why mathematics matters. It is not a vehicle-design, repair or compliance guide. A real windshield is curved, the rubber blade deforms, the arm has pressure and articulation, and the wiped region is evaluated under product-specific and legal requirements. As one concrete example, the United States rule at 49 CFR §571.104 specifies requirements for windshield wiping and washing systems, including wiped-area and frequency provisions for covered vehicles. That is a jurisdiction-specific rule, not a universal standard and not evidence that a classroom sketch certifies any vehicle.
The useful habit is to separate four questions: what path does the mechanism command, what area does the blade sweep, how quickly does it move, and how well does that model represent the real surface?
A Reading Route
- Start with annular sectors for swept-area geometry.
- Convert angles into distances to see why the outer tip moves farther.
- Study the linkage to connect coordinates with mechanisms.
- Compare two blades using overlap and set unions.
- Follow the audit to build a careful model without touching a real mechanism.
The First Model: An Annular Sector
Imagine a straight blade extending from inner radius (r) to outer radius (R) from a fixed pivot. If it rotates through angle ( heta) degrees on a flat plane, the ideal swept area is the difference between two circular sectors:
(A=\frac{\theta}{360^\circ}\pi(R^2-r^2)).
This is an annular sector. “Annular” means ring-shaped. The inner radius matters because a real blade does not normally begin at the exact pivot.
Worked Example
Let (R=0.55) m, (r=0.10) m and ( heta=100^\circ). Then
(R^2-r^2=0.55^2-0.10^2=0.3025-0.01=0.2925\text{ m}^2).
So
(A=\frac{100}{360}\pi(0.2925)\approx0.2553\text{ m}^2).
This value is the area of the ideal flat annular sector. It is not automatically the true surface area wiped on curved glass.
Reasonableness Checks
- If ( heta=0^\circ), area is zero.
- If ( heta=360^\circ), area is the full annulus (pi(R^2-r^2)).
- Increasing (R) increases area through (R^2).
- Increasing (r) while holding everything else fixed reduces area.
These limiting cases test the structure of the formula. A result that fails them is not credible.
Why Blade Length Alone Is Not Enough
Two blades can have equal length (R-r) but different swept areas if they sit at different distances from their pivots. A blade from 0.10 m to 0.50 m and a blade from 0.30 m to 0.70 m both have length 0.40 m. Yet the ring terms are
(0.50^2-0.10^2=0.24)
and
(0.70^2-0.30^2=0.40).
At the same angle, the second ideal sector is larger. Position relative to the pivot matters, not only physical blade length.
Angle, Arc Length and Tip Motion
An angle can be measured in degrees or radians. Arc length uses radians naturally:
(s=r\theta),
where ( heta) is in radians. Convert degrees by multiplying by (pi/180).
For a 100-degree sweep,
( heta=100\pi/180\approx1.7453\text{ rad}).
At outer radius 0.55 m, the tip travels one-way arc length
(s=0.55\times1.7453\approx0.960\text{ m}).
At inner radius 0.10 m, a point travels only about 0.175 m over the same angle. Every point has the same angular displacement, but linear distance increases with radius.
One Cycle Must Be Defined
In many contexts, one wiping cycle means movement from one extreme to the other and back. If the one-way sweep angle is ( heta), the outer tip travels about (2R\theta) per ideal cycle along its arc.
Using (R=0.55) m and ( heta=1.7453) rad gives
(2\times0.55\times1.7453\approx1.920\text{ m per cycle}).
If someone instead calls each one-way movement a “cycle,” the calculated frequency and path distance will differ by a factor of two. Define the word before calculating.
Cycles per Minute and Period
Forty-five cycles per minute equals
(45/60=0.75\text{ cycles per second}=0.75\text{ Hz}).
The period is the reciprocal:
(T=1/0.75\approx1.333\text{ s per cycle}).
The cited US rule includes a high-frequency condition of at least 45 cycles per minute and a lower frequency of at least 20 cycles per minute with a specified difference. Those are legal provisions in that rule's scope, not a general instruction for testing or modifying a vehicle.
Average Tip Path Speed
If the ideal tip travels 1.920 m per cycle at 45 cycles per minute, path distance per minute is
(1.920\times45=86.4\text{ m/min}),
or 1.44 m/s on average while treating the full cycle time as motion. Real motion accelerates, decelerates and may include dynamic effects. The average does not state the instantaneous maximum.
Angular Speed and Linear Speed Are Related but Different
For angular speed (omega) in radians per second, the tangential speed of a point at radius (r) is
(v=\omega r).
At the same instant, points on one rigid arm have the same (omega), but the outer point has greater (v). If (omega=2) rad/s:
- at 0.10 m, (v=0.20) m/s;
- at 0.30 m, (v=0.60) m/s;
- at 0.55 m, (v=1.10) m/s.
This difference is why “the blade speed” is incomplete unless the location is named. The inner and outer ends travel different linear distances in the same time.
Acceleration Appears at Reversals
The blade must reverse direction at each end of its sweep. Its angular velocity passes through zero and changes sign. A constant-speed sector drawing does not show this. A time graph of angle resembles a smooth oscillation more than a triangle with perfectly sharp corners, because real mechanisms cannot change velocity instantaneously.
For a simple sinusoidal classroom model,
(phi(t)=\phi_0+a\sin(2\pi ft)).
Then angular velocity is
(omega(t)=2\pi fa\cos(2\pi ft)).
The model predicts greatest speed near the middle and zero speed at endpoints. A real linkage may produce an asymmetric profile, but the sinusoid is a useful first comparison.
Distance, Displacement and Coverage
A full out-and-back cycle has zero net angular displacement because the blade returns to its starting angle. Yet it travels a positive path distance and sweeps area twice. This distinction between displacement and distance is fundamental in mechanics.
Why a Linkage Does Not Sweep Uniformly
A motor can rotate continuously while the wiper arm oscillates. A crank-rocker linkage converts rotation into back-and-forth motion. The components can be idealised as rigid bars connected by rotating joints.
Let the motor crank endpoint be
(P=(a\cos\alpha,a\sin\alpha)),
where (a) is crank length and (alpha) is input angle. Another joint must lie at the intersection of circles determined by the link lengths and fixed pivots. Solving those circle intersections yields the rocker position and output angle.
Coordinate Construction
A student model can proceed without servicing anything:
- place the motor pivot at ((0,0));
- place the rocker pivot at ((d,0));
- choose fictional link lengths that form a movable four-bar;
- calculate the crank endpoint for several input angles;
- find the coupler-rocker joint geometrically;
- record the output rocker angle.
Plot input angle against output angle. The graph will usually not be a straight line. Equal motor-angle steps can produce unequal wiper-angle steps.
Transmission and Sensitivity
The slope (d\phi/d\alpha) describes how output angle changes with input angle locally. A large magnitude means a small motor rotation creates a larger rocker change. Near certain configurations, sensitivity can change rapidly.
This is one reason simplistic calculations based only on average cycle rate cannot describe instantaneous blade speed. Geometry shapes the timing.
Multiple Mathematical Assemblies
Circle intersections can produce more than one possible joint position. A model must choose the assembly corresponding to the intended configuration and maintain it across steps. Jumping between branches creates an impossible motion. This is an important general lesson for equation solving: a mathematical solution may satisfy equations while violating continuity or physical constraints.
Linkage Limits
Real components have thickness, joint clearance, friction, flexible elements and packaged obstacles. A four-line diagram omits all of these. Use it to explain kinematic relationships, not to specify parts or modifications.
The Swept Mask Is More Useful Than One Number
Area alone does not show where wiping occurs. A mask represents the set of points reached by the ideal blade. Overlaying it on a windshield outline shows top gaps, corner gaps and overlap.
Coordinate Test for a Sector
Relative to pivot ((x_0,y_0)), a point ((x,y)) lies within an ideal annular sector when:
- its distance (
- =\sqrt{(x-x_0)^2+(y-y_0)^2}) satisfies (r\le\rho\le R);
- its polar angle lies within the declared angular interval.
A grid program or spreadsheet can test each cell centre. Counting accepted cells times cell area approximates the swept area.
Resolution Error
Boundary cells are only partly covered. A coarse 5 cm grid can misclassify much more area than a 1 cm grid. Recalculate at several resolutions. If estimates converge, confidence improves. The difference between resolutions is evidence about numerical uncertainty.
Priority Region
Not all points necessarily have equal value in a design question. A simplified priority viewing region might be a declared rectangle or polygon. Coverage percentage is
(100\times\frac{|W\cap P|}{|P|}%),
where (W) is the wiped mask and (P) is the priority region. This metric is only as meaningful as the chosen region. A classroom priority mask is not a legal visibility definition.
Boundary Clearance
At sampled angles, calculate blade endpoint positions and their distance to a declared border. The smallest sampled distance is not necessarily the true minimum between samples. Finer sampling or analytic optimisation may be needed. Never use an informal classroom calculation as a clearance certification.
Two Wipers Create a Set Union
Let the two ideal masks be (W_1) and (W_2). Total unique wiped area is
(|W_1\cup W_2|=|W_1|+|W_2|-|W_1\cap W_2|).
The intersection is counted in both individual areas, so subtract it once. This is the inclusion-exclusion principle.
Overlap Can Serve Different Roles
Some overlap may help avoid a central unwiped seam as blades move. Too much overlap could repeat coverage without adding unique area or create interference risk in a physical system. The classroom model should not label all overlap good or bad. It should measure:
- unique total area;
- shared area;
- uncovered priority area;
- minimum distance between idealised blades at synchronised time steps.
The last measure needs motion timing, not just static masks. Two blades can occupy the same region at different times without physically colliding.
Synchronisation as Phase
Represent the motions as (phi_1(t)) and (phi_2(t)). If the second motion is shifted by phase (delta), then
(phi_2(t)=\phi_1(t+\delta))
in a simplified identical-motion model. Changing phase alters simultaneous clearance but not each blade's individual swept mask. This shows why spatial coverage and temporal coordination are separate mathematical layers.
Parking Positions
An angular park error (Delta\theta) creates approximate outer-tip displacement (R\Delta\theta) when the angle is in radians. At (R=0.55) m, a one-degree error gives
(0.55\times\pi/180\approx0.00960\text{ m}=9.60\text{ mm}).
A two-degree error gives about 19.2 mm. Small angular changes can create visible endpoint movement at a long radius.
Curved Glass Changes the Meaning of Area
The annular-sector formula is planar. A windshield is a three-dimensional surface. Projecting a blade path onto a front-view drawing can be useful for visible coverage, but the drawing's area is not necessarily the glass surface area.
Projection Distorts Length and Area
A tilted surface appears foreshortened. A curve on the glass can project to a shorter line in the image plane. If the local surface is tilted by angle (eta) relative to a viewing plane, a simple orthographic projection can scale one component by (cos\beta). Real curvature varies across the glass, so one factor is not enough.
Blade Conformity
The blade must contact a changing surface. A straight line in one view does not prove full contact. Flexible blade structure, arm force and material behaviour lie beyond the rigid planar sector model.
Which Area Should Be Reported?
A study might report:
- projected area in a front-view plane;
- parameterised surface area on a 3D model;
- measured mask area from an image under a declared camera geometry.
These are different. A strong report names the representation rather than calling every result simply “wiped area.”
Tolerance Turns One Answer into a Range
Nominal dimensions are rarely exact. Let outer radius, inner radius and angle vary within fictional tolerances. Calculate area at combinations of extremes or use repeated simulation.
For
(A=\frac{\theta}{360}\pi(R^2-r^2)),
area increases with ( heta) and (R) and decreases with (r). A conservative maximum uses the largest ( heta), largest (R) and smallest (r); a conservative minimum uses the opposite extremes, if variables can vary independently.
Sensitivity to Outer Radius
The derivative with respect to (R) is
(\frac{\partial A}{\partial R}=\frac{\theta}{180}\pi R)
when ( heta) is in degrees in this expression. The result grows with (R), so the same small radial change affects area more at a larger radius.
Correlated Variables
Worst-case combinations may be impossible if variables are linked by geometry or manufacturing. A simulation that samples every variable independently can therefore exaggerate or misrepresent variation. Record correlation assumptions.
Tolerance Is Not Failure
If a measured value differs slightly from nominal but falls within a declared valid tolerance, that is not automatically an error. Whether a tolerance is acceptable depends on the actual engineering and regulatory context. Classroom values are illustrative only.
Intermittent Timing Is a Duty-Cycle Problem
An intermittent mode alternates sweep periods with pauses. Suppose one complete sweep cycle lasts 1.4 s and the following pause lasts 4.6 s. The repeating interval is 6.0 s, so the long-run average is
(60/6=10\text{ cycles per minute}).
The mechanism's speed during the 1.4 s motion is not the same as the average over motion plus pause.
Duty Fraction
The fraction of time in motion is
(1.4/6.0\approx0.233), or 23.3%.
This ratio appears in electronics, pumps, heating controls and communication systems. The mathematics transfers because each system alternates active and inactive intervals.
Variable Intervals
If pause durations change, calculate a time-weighted average over the full observation. Do not average displayed rates unless they cover equal durations. Keep the original time series so bursts and long gaps are visible.
Sampling a Fast Motion
A low-frame-rate video may miss endpoints or maximum speed. If sampling frequency is too low relative to motion, aliasing can make direction and timing appear wrong. This links wiper analysis with digital signal processing.
A Reproducible Student Design Audit
Step One: Create a Fictional Coordinate System
Draw a windshield boundary and one or two pivot points on graph paper or in geometry software. Declare that the dimensions are invented and not copied from a vehicle.
Step Two: Define Each Blade
Record inner radius, outer radius, start angle and end angle. State whether the model treats the blade as a radial line and the surface as flat.
Step Three: Calculate Analytic Areas
Use annular-sector formulas for each independent mask. Check units and limiting cases.
Step Four: Rasterise the Masks
Test grid cells against distance and angle conditions. Form the union and intersection. Repeat with a finer grid and compare.
Step Five: Add Motion
Choose a declared angle-time function or a fictional linkage. Define one cycle. Convert cycles per minute to hertz and period. Plot angle and angular speed.
Step Six: Add Uncertainty
Vary pivot location, radii and angles within fictional tolerances. Report ranges for area, gap and overlap.
Step Seven: State the Boundary
Explain that the result is a mathematical exercise, not a safety assessment, legal conclusion, repair procedure or instruction to alter a vehicle.
Building the Geometry in a Spreadsheet or Program
A reproducible numerical model can be built from a list of grid-point coordinates. The method is simple enough for a spreadsheet yet rich enough to teach Boolean logic and coordinate transformations.
For every grid point, subtract the pivot coordinates:
(x'=x-x_0\), (y'=y-y_0).
Calculate radius
(\rho=\sqrt{x'^2+y'^2})
and direction
(\psi=\operatorname{atan2}(y',x')).
The two-argument atan2 function is preferable to ordinary inverse tangent because it identifies the correct quadrant. Convert its output consistently to degrees or keep all angles in radians.
Handling an Angle That Crosses Zero
Suppose the swept interval runs from 330 degrees to 40 degrees. The naive test (330\le\psi\le40) can never be true. Normalise angles to 0–360 degrees and use
(psi\ge330^\circ\text{ or }\psi\le40^\circ).
This wrap-around issue appears in compass bearings, clocks and phase signals. Correct interval logic is part of the mathematics, not merely a programming detail.
Boolean Masks
For each point, define:
- radial test: (r\le\rho\le R);
- angular test: (psi) is within the swept interval;
- windshield test: the point is inside the declared boundary.
The ideal wiped mask is the logical AND of all three. For two blades, union is OR and overlap is AND. Counting true cells and multiplying by cell area produces numerical area estimates.
Cell Centres and Boundary Cells
Testing cell centres is easy but imperfect. A curved boundary can cut a cell even when its centre lies outside. Subdivide boundary cells or compare multiple grid sizes. Report the rule used; otherwise another student may get a different answer from the same drawing.
A Numerical Cross-Check
Create one unbounded annular sector whose analytic area is known. If the grid estimate differs by 8% on a coarse grid and 2% on a finer grid, the direction of improvement is encouraging. If it does not converge, inspect angle units, cell area and wrap-around logic.
Comparing Coverage Across Designs Fairly
A larger nominal swept area does not automatically mean better priority coverage. Consider two fictional designs on the same projected window:
- Design A sweeps 0.72 square metres, of which 0.40 square metres lies in a 0.45-square-metre priority region.
- Design B sweeps 0.65 square metres, of which 0.42 square metres lies in that priority region.
Design A has more total area. Its priority coverage is (0.40/0.45\approx88.9\%). Design B covers (0.42/0.45\approx93.3\%). According to priority coverage, B performs better despite the smaller total mask.
Coverage Efficiency
One optional descriptive ratio is useful priority area divided by total swept area. In the example:
- A: (0.40/0.72\approx55.6\%);
- B: (0.42/0.65\approx64.6\%).
This ratio rewards concentration in the declared priority region. It should not replace other needs, such as broader visibility, drainage or boundary clearance. It simply answers one defined question.
Gap Size Matters as Well as Gap Area
Two designs can leave the same uncovered area in different shapes. One may leave a thin border strip; another may leave a compact gap in a critical location. Report maximum gap dimensions or distance-to-covered-region maps when shape matters. A single percentage can hide topology and location.
Multi-Objective Comparison
A comparison table can include priority coverage, total coverage, overlap, maximum gap, clearance and sensitivity to tolerance. Avoid collapsing everything into one weighted score unless the weights are declared and tested. Often a Pareto comparison—showing which designs trade one benefit for another—is more informative.
Do Not Optimise Past the Model
A numerical optimiser can move pivots and angles until the flat mask looks excellent. If the model omits curvature, packaging, blade contact and legal definitions, the apparent optimum may be meaningless. Optimisation magnifies modelling assumptions; it does not repair them.
Comparing Uncertainty Bands
Instead of reporting only one coverage percentage, repeat the calculation across plausible fictional tolerances. If Design A gives 91% to 95% priority coverage and Design B gives 92% to 93%, A has the higher nominal possibility but B is less sensitive in this model. Whether robustness is preferred depends on the declared objective.
Traceability of Inputs
Store pivot coordinates, radii, angle conventions, grid size, priority mask and tolerance ranges beside the result. A screenshot alone is not enough: another student should be able to reconstruct every cell test. Traceability is part of mathematical communication because a number without its definitions cannot be independently checked.
One-at-a-Time Sensitivity
Change outer radius, sweep angle and pivot position separately. Record the change in unique coverage and maximum gap. This reveals local influence, though it can miss interactions between variables. A second experiment can change pairs together to test whether their combined effect differs from adding the separate effects.
Separate Geometric and Timing Results
The swept union answers where a blade can reach over a complete motion. A synchronised time simulation answers where both blades are at a particular instant. Do not use the static union to infer simultaneous clearance. Store time step, angle law and phase convention with every animation or distance calculation.
Check Symmetry Before Assuming It
Two pivots placed symmetrically do not guarantee symmetric coverage if radii, angles or timing differ. Reflect one mask across the chosen centreline and compare it cell by cell with the other. The count of mismatched cells measures symmetry in that numerical representation. A visually balanced diagram can still have measurable differences.
Visualisation Should Preserve Scale
Use equal axis scaling so one metre horizontally occupies the same plotted length as one metre vertically. Otherwise circles appear as ellipses and clearances can look misleading. Include the coordinate origin, units and legend on every coverage plot.
Common Misconceptions and Better Questions
“Every Point on the Blade Moves at the Same Speed”
Every point shares angular speed in a rigid rotation, but (v=\omega r) means linear speed grows with radius. Ask: speed at which point?
“One Cycle Is One Sweep”
Usage varies. Ask: does the definition mean one-way or out-and-back?
“Area Equals Visibility”
Location matters. Ask: how much of the declared priority region is covered?
“A Constant-Speed Motor Means Constant-Speed Wiping”
A linkage can map equal input rotations to unequal output-angle changes. Ask: what is the input-output relationship?
“The Flat Sector Is the True Glass Area”
Curvature and projection matter. Ask: is the value planar, projected or surface area?
“Meeting One Frequency Number Proves Compliance”
Compliance involves the full applicable rule, test procedures, vehicle category and qualified evidence. Ask: which jurisdiction, edition, scope and official assessment apply?
Guidance for Students, Parents and Teachers
For Students
Build a fictional diagram. Do not measure moving vehicle components or attempt adjustments. Show degrees-to-radians conversion explicitly and define a cycle before calculating rate.
For Parents
Ask why the outer tip travels farther than the inner end. A paper strip rotating around a drawing pin can demonstrate radius and arc length safely away from any vehicle.
For Teachers
The topic joins sectors, coordinate geometry, sets, rates, trigonometry and numerical methods. Use legal or technical documents only to show how precise definitions matter; do not ask students to certify compliance.
| Model layer | Main quantities | Main limitation |
|---|---|---|
| Sector | (r,R,\theta,A) | Flat rigid geometry |
| Motion | angle, period, speed | Simplified time law |
| Linkage | coordinates, bar lengths | Ideal joints and bars |
| Coverage | union, overlap, gaps | Chosen projection and grid |
| Tolerance | ranges, sensitivity | Assumed distributions |
Frequently Asked Questions
What shape does one ideal wiper blade sweep?
On a flat model with fixed pivot and radial blade, it sweeps an annular sector.
Why subtract squared radii?
The swept region is an outer sector minus an inner sector. Circle and sector areas depend on radius squared.
How far does the blade tip travel?
For one-way angular sweep ( heta) radians at radius (R), arc length is (R\theta). An out-and-back ideal cycle travels approximately (2R\theta).
Does a longer blade always produce better coverage?
It can increase nominal swept area, but boundaries, curvature, pressure, linkage geometry, overlap and applicable requirements matter. “Better” needs a declared metric and constraints.
Why use a grid if there is a formula?
The formula works for a clean annular sector. A grid helps estimate clipped, overlapping or irregular masks. Comparing resolutions reveals approximation error.
Can students test a real vehicle?
This article does not provide a vehicle test or adjustment procedure. Use fictional drawings, safe classroom mechanisms or public datasets, and leave real inspection and service to applicable official procedures and qualified professionals.
Next Reading and a Final Challenge
Linkages also explain drawing mechanisms. Read Why Mathematics? | Pantographs, Scale Factors and Linkage Geometry. For another moving-coverage problem, see Why Mathematics? | Robot Vacuums, Occupancy Grids and Coverage Paths. For motion measured from images, continue to Why Mathematics? | Optical Flow, Image Gradients and Motion Estimation. More connections are collected in the Mathematics Learning Hub.
For a final challenge, design two fictional wipers on a 1.4 m by 0.7 m rectangular coordinate window. Give each an inner radius, outer radius and angular interval. Estimate individual areas analytically, calculate union and overlap on two grid resolutions, and measure coverage of a smaller priority rectangle. Then introduce a one-degree park-angle tolerance and report how far each outer tip can move.
The strongest answer will not merely maximise one number. It will explain what each number represents, what the flat model omits, and why a reproducible calculation is more valuable than a confident sketch.
A Practical Mathematics Studio
These investigations use drawings, synthetic data or ordinary supervised operation within a manufacturer's instructions. They are learning activities, not installation, compliance, maintenance or repair instructions. Record assumptions and make every result reproducible.
Investigation 1: Draw an annular sector
Choose inner and outer radii plus sweep angle, then calculate theta/360 times pi times the difference of squared radii. Verify limiting cases at zero and 360 degrees.
Investigation 2: Compare blade lengths
Keep pivot and angle fixed while increasing outer radius. Calculate the area change and explain why it depends on squared radii, not just extra blade length.
Investigation 3: Track tip speed
For angular speed omega and radius r, calculate v = omega r at several points along the blade. The outer tip travels faster than a point near the pivot.
Investigation 4: Convert cycles per minute
Translate 45 cycles per minute into cycles per second and period per cycle. Define whether one cycle means out-and-back before calculating angular travel.
Investigation 5: Build a crank-rocker sketch
Choose four linkage lengths, draw several crank positions, and locate the rocker angle geometrically. Treat the model as illustrative, not a vehicle design.
Investigation 6: Make a coupler table
Record input crank angle, joint coordinates and output angle. Plot the mapping to see why uniform motor rotation need not create uniform blade angle.
Investigation 7: Estimate useful coverage
Overlay a priority viewing region and a swept mask on graph paper. Count intersection cells and divide by priority-region cells, documenting boundary treatment.
Investigation 8: Find overlap between blades
Model two swept masks and count shared area. Separate useful continuity from unnecessary repeated wiping instead of assuming all overlap is waste.
Investigation 9: Check pillar clearance
Add a windshield boundary and calculate the minimum distance between blade endpoints and the boundary across sampled angles. Sampling can miss the true minimum, so state resolution.
Investigation 10: Model parking position
Compare angular park errors of one and two degrees at the outer tip using arc length s = r theta with theta in radians.
Investigation 11: Introduce tolerance
Vary pivot location, blade length and sweep angle within fictional tolerances. Report the range of swept areas rather than only the nominal value.
Investigation 12: Compare flat and curved glass
Project the same angular sweep onto a flat diagram and note what the model omits about a curved glazing surface. Avoid treating plan area as true surface area.
Investigation 13: Audit intermittent timing
Create a time line with sweep duration and pause duration. Calculate average cycles per minute and distinguish it from motor speed during motion.
Investigation 14: Test sampling error
Estimate area on coarse and fine grids. Compare results and use the difference as evidence about numerical resolution.
Investigation 15: Separate compliance and design
Read an official wiping-system rule, identify what it specifies, and distinguish that requirement from your simplified geometry. Do not certify any vehicle.
Investigation 16: Prepare a design report
Include coordinate system, linkage assumptions, blade radii, angular range, coverage mask, cycle definition, tolerances and limitations so a peer can reproduce the calculation.